Related papers: On Achievability of an $(r,l)$ Fractional Linear N…

For any field k and any integers m,n with 0 <= 2m <= n+1, let W_n be the k-vector space of sequences (x_0,...,x_n), and let H_m be the subset of W_n consisting of the sequences that satisfy a degree-m linear recursion, that is, for which…

We consider an arbitrary layered Gaussian relay network with $L$ layers of $N$ relays each, from which we select subnetworks with $K$ relays per layer. We prove that: (i) For arbitrary $L, N$ and $K = 1$, there always exists a subnetwork…

In this work we study the existence of positive solution to the fractional quasilinear problem, $$ \left\{ \begin{array}{rcll} (-\Delta )^s u &=&\lambda \dfrac{u}{|x|^{2s}}+ |\nabla u|^{p}+ \mu f &\inn \Omega,\\ u&>&0 & \inn\Omega,\\ u&=&0…

For positive integers m and r, one can easily show there exist integers N such that for every map D:{1,2,...,N} -> {1,2,...,r} there exist 2m integers x_1 < ... < x_m < y_1 < ... < y_m which satisfy: (a) D(x_1) = ... = D(x_m), (b) D(y_1) =…

In many real complex networks, the fractal and self-similarity properties have been found. The fractal dimension is a useful method to describe fractal property of complex networks. Fractal analysis is inadequate if only taking one fractal…

In this article, we propose the fractional weak adversarial networks (f-WANs) for the stationary fractional advection dispersion equations (FADE) based on their weak formulas. This enables us to handle less regular solutions for the…

Family of equations, which is the generalization of the $K(m,m)$ equation, is considered. Periodic wave solutions for the family of nonlinear equations are constructed.

Fractional calculus is a powerful and effective tool for modelling nonlinear systems. The M derivative is the generalization of alternative fractional derivative. This M derivative obey the properties of integer calculus. In this paper, we…

In this paper we consider a linear homogeneous system of $m$ equations in $n$ unknowns with integer coefficients over the reals. Assume that the sum of the absolute values of the coefficients of each equation does not exceed $k+1$ for some…

We consider directed acyclic {\em sum-networks} with $m$ sources and $n$ terminals where the sources generate symbols from an arbitrary alphabet field $F$, and the terminals need to recover the sum of the sources over $F$. We show that for…

We consider the question of global existence of small, smooth, and localized solutions of a certain fractional semilinear cubic NLS in one dimension, $$i\partial_t u - \Lambda u = c_0{|u|}^2 u + c_1 u^3 + c_2 u \bar{u}^2 + c_3 \bar{u}^3,…

For positive integers $m$ and $n$, the Zarankiewicz number $Z_{2,2}(m,n)$ can be defined as the maximum total degree of a linear hypergraph with $m$ vertices and $n$ edges. Guy determined $Z_{2,2}(m,n)$ for all $n \geq \binom{m}{2}/3+O(m)$.…

Circular-shift linear network coding (LNC) is a class of vector LNC with low encoding and decoding complexities, and with local encoding kernels chosen from cyclic permutation matrices. When $L$ is a prime with primitive root $2$, it was…

For k>=3 let A \subset [1,N] be a set not containing a solution to a_1 x_1+...+a_k x_k=a_1 x_{k+1}+...+a_k x_{2k} in distinct integers. We prove that there is an epsilon>0 depending on the coefficients of the equation such that every such A…

In these notes, we describe the strategy for the derivation of the hydrodynamic limit for a family of long range interacting particle systems of exclusion type with symmetric rates. For $m \in \mathbb{N}:=\{1, 2, \ldots\}$ fixed, the…

Let $A$ be a random $m\times n$ matrix over the finite field $F_q$ with precisely $k$ non-zero entries per row and let $y\in F_q^m$ be a random vector chosen independently of $A$. We identify the threshold $m/n$ up to which the linear…

You have $m$ muffins and $s$ students. You want to divide the muffins into pieces and give the shares to students such that every student has $\frac{m}{s}$ muffins. Find a divide-and-distribute protocol that maximizes the minimum piece. Let…

We study scalar-linear and vector-linear solutions to the generalized combination network. We derive new upper and lower bounds on the maximum number of nodes in the middle layer, depending on the network parameters. These bounds improve…

This paper is devoted to the study of meromorphic solutions of nonlinear differential equations, specifically the equation \[ (f^n)^{(k)}(g^n)^{(k)} = \alpha^2, \] where $k$ and $n$ are positive integers with $n>2k$, and $\alpha$ is a…

Complex networks have recently attracted much attention in diverse areas of science and technology. Many networks such as the WWW and biological networks are known to display spatial heterogeneity which can be characterized by their fractal…